# An Entropy Analysis of Classroom Conditions Based on Mathematical Social Science

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## Abstract

**:**

## 1. Introduction

## 2. Methods: Entropy in a Sociological Model of Opinion Decision-Making Processes

#### 2.1. Entropy in the Information Theory

_{B}logW, which means the disorder of constituents in the system, where W is the total number of possible arrangements of particles and k

_{B}is the Boltzmann constant. The increase in entropy S corresponds to the increase in disturbance of the condition of the system.

_{1}, …, p

_{m}) is called “entropy of probability distribution”. We can see that the concept of entropy in the information theory includes the concept in statistical physics. If p

_{i}= 1/W, Boltzmann’s entropy is the same as Shannon’s entropy, except for a multiple constant, as shown in the following.

**R**is a set of real numbers. The differential entropy depends only on the probability density of random variables. However, there are some differences between the continuous and the discrete cases. The most important point is that the differential entropy H

_{d}(X) can take negative values, whereas the entropy H(X) in the discrete case always satisfies H(X) > 0. Although the differential entropy can have not only positive value but also a negative value, it is known that the difference H

_{d}(X)-H

_{d}(Y) is a meaningful quantity in the system [14]. That is, the difference H

_{d}(X)-H

_{d}(Y) means the increase or decrease in disturbance in the system.

#### 2.2. The Weidlich Model for the Stochastic Process on the Change of Opinions

_{st}(x). The parameters k and h, on the other hand, are determined by the parameter fitting, comparing the results of numerical calculations with the observed values μ

_{x}(the mean value of x) and σ

_{x}

^{2}(variance of x). Then, the stationary probability distribution f

_{st}(x) is obtained with the values of k and h, as shown in Figure 4. The details of calculation are described in Appendix A.

#### 2.3. Differential Entropy of Stationary Distribution on the Weidlich Model

_{d}(k, h) on the stationary distribution f

_{st}(x; k, h), which is determined by two parameters of k and h in the Weidlich model, as follows.

## 3. Materials

#### 3.1. Data Obtained from Classroom

- “Did you calmly spend in every lesson today?”
- “Did you do good greetings of the beginning and end in every lesson today?”
- “Did you eat all the lunch today?”
- “Did you do cleaning earnestly today?”
- “Did you participate positively in the Home-Room time today?”

#### 3.2. Setting of Two-States Variables on the Desk Arrangements and the Questionnaires

## 4. Results: Calculations of (k, h) and H_{d} (k, h) with the Obtained Data

_{d}(k, h) by using the data of the desk arrangements and the questionnaire about attitudes in lessons obtained in Section 3.

_{d}(k, h) over the three months for the classes (a), (b) and (c).

## 5. Discussion: Classroom Conditions and Calculated Results

_{d}(k, h) by using the data of the desk arrangements and the questionnaire on attitudes in lessons, where the parameter k is a measure of the strength of adaptation to neighbors and h is a preference parameter. One should note that the parameter h can be interpreted as a “moral parameter”, as all the questions in the questionnaire and the desk arrangements are related to students’ morals. Here, we discuss the relation between the classroom conditions and the calculated results.

_{d}(k, h), which are numerically calculated by using the data of the desk arrangements and the questionnaire on attitudes in lessons. A remarkable change can be found in the parameters (k, h) and H

_{d}of Class (b) during September to December. It shows that k (a measure of the strength of adaptation to neighbors) increases but h (a moral parameter) slightly decreases, and differential entropy H

_{d}(mental disorder of the group) increases, in both investigations. This means that the interactions of each member were strong, but their morals were slightly weak, and the mental stability of the group was disturbed.

_{d}increase but h slightly decreases. The increase of parameter k means that the interaction force of students becomes strong. The increase in differential entropy H

_{d}can be interpreted as the increase in the instability of the group. The decrease in parameter h, on the other hand, means that the morale of students becomes weaker. In this condition, if some negative feelings accidently happen, they will appear as fluctuations, but these should be enlarged by both the increase in k (interaction force of students) and H

_{d}(mental disorder of the group). Thus, this situation matches the occurrence of bullying.

_{d}may be a symptom of bullying. This is similar to a totalitarian society. Thus, in this study, we see some similarities between the classroom and society.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Determination of Parameters in Weidlich Model

_{st}(x) of the Fokker-Planck equation is calculated, giving:

_{st}(x), ν is not included in the result of K

_{1}(y)/K

_{2}(y), hence important parameters for stationary states are k and h.

_{x}and σ

_{x}

^{2}are the mean value of x and variance in measurements respectively. By comparing the results of numerical calculations with the observed values μ

_{x}and σ

_{x}

^{2}, k and h can be determined. Here we emphasize that parameters k and h can be obtained by the statistical numerical calculations on the target human group, hence these parameters should reflect the state of the group.

## Appendix B. Behaviors of the Differential Entropy on the Weidlich Model

_{st}(x), where the preference parameter of h are all fixed to zero (h = 0) but the measure of the strength of adaptation to neighbors (k) increases gradually. The values of the parameters of k, h, μ

_{x}, σ

_{x}

^{2}, and H

_{d}(k, h) are shown at the bottom of Figure A1. All the mean values μ

_{x}are zero as h = 0. The value of differential entropy H

_{d}(k, h) varies not only downwards but also upwards. So, H

_{d}(k, h) does not always increase with increasing the value of k. Thus, the behavior of entropy on the Weidlich model is seen to be not simple.

**Figure A1.**Probability distributions f

_{st}(x) with symmetric features against y axis. Related parameters and differential entropy are also shown.

_{st}(x), whose parameters h have opposite sign each other. The values of the parameters of k, h, μ

_{x}, σ

_{x}

^{2}, and H

_{d}(k, h) are shown at the bottom of Figure A2. These graphs have the property of mirror symmetry against y-axis, so that they have the same variance and entropy.

**Figure A2.**Probability distributions with opposite sign of h. Related parameters and differential entropy are also shown.

_{x}takes the same value. The values of the parameters of k, h, μ

_{x}, σ

_{x}

^{2}, and H

_{d}(k, h) are shown at the bottom of Figure A3. Comparing the varying of variance σ

_{x}

^{2}and the differential entropy H

_{d}(k, h), we can find the variance increases but the differential entropy decreases. That is, the increase of the variance of probability distribution does not always correspond to the increase of entropy. So, the differential entropy and the variance of probability distribution show different behavior, and they should have different meaning.

**Figure A3.**Probability distributions with different behavior but with the same mean value. Related parameters are also shown. The variance increases but the differential entropy decreases in spite of the same mean value of f

_{st}(x).

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**Figure 2.**Weidlich model in “Synegetics. In the framework of synergetics for interdisciplinary realms, the Weidlich model can describe the human decision-making process by an analogy with the ferromagnetic spin model.

**Figure 3.**Spin model in statistical physics. The spin model is constructed by two kinds of magnetic dipole, called spin up ↑ and down ↓ (opposite direction).

**Figure 5.**Pictures for (

**a**) lunch time and (

**b**) a few minutes later, after the rearrangement of desks.

**Figure 7.**Changes in the parameters (k, h) and differential entropy H

_{d}(k, h) on the desk arrangements and the attitudes in lesson times.

**Table 1.**Calculated results of parameters and differential entropy on the desk arrangements and attitudes in lesson times.

Class (a) | Class (b) | Class (c) | |||||||
---|---|---|---|---|---|---|---|---|---|

Month | Jun. | Sep. | Dec. | Jun. | Sep. | Dec. | Jun. | Sep. | Dec. |

${\mu}_{x}$: desk | 0.140 | 0.460 | 0.405 | 0.115 | 0.075 | 0.310 | 0.195 | 0.295 | 0.259 |

${\mu}_{x}$: attitudes | −0.096 | 0.027 | 0.143 | −0.022 | 0.046 | 0.068 | −0.063 | −0.011 | −0.0014 |

${{\sigma}_{x}}^{2}$: desk | 0.0108 | 0.0014 | 0.0051 | 0.0171 | 0.0034 | 0.0133 | 0.0226 | 0.0171 | 0.0219 |

${{\sigma}_{x}}^{2}$: attitudes | 0.0126 | 0.0260 | 0.0247 | 0.0387 | 0.0079 | 0.0310 | 0.0379 | 0.0160 | 0.0064 |

$k$: desk | 1.06 | 3.46 | 2.68 | 1.45 | −1.69 | 2.16 | 1.88 | 2.17 | 2.08 |

$k$: attitudes | 1.14 | 1.69 | 1.79 | 1.93 | 0.46 | 1.82 | 1.94 | 1.30 | 0.012 |

$h$: desk | 0.147 | 0.112 | 0.109 | 0.079 | 0.277 | 0.110 | 0.079 | 0.095 | 0.085 |

$h$: attitudes | −0.091 | 0.013 | 0.062 | −0.007 | 0.072 | 0.026 | −0.02 | −0.009 | −0.003 |

${H}_{d}$: desk | −0.850 | −2.258 | −1.485 | −0.638 | −1.428 | −0.902 | −0.549 | −0.794 | −0.651 |

${H}_{d}$: attitudes | −0.772 | −0.413 | −0.478 | −0.243 | −0.999 | −0.348 | −0.262 | −0.638 | −1.103 |

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Koyama, K.; Niwase, K. An Entropy Analysis of Classroom Conditions Based on Mathematical Social Science. *Educ. Sci.* **2019**, *9*, 288.
https://doi.org/10.3390/educsci9040288

**AMA Style**

Koyama K, Niwase K. An Entropy Analysis of Classroom Conditions Based on Mathematical Social Science. *Education Sciences*. 2019; 9(4):288.
https://doi.org/10.3390/educsci9040288

**Chicago/Turabian Style**

Koyama, Kazuo, and Keisuke Niwase. 2019. "An Entropy Analysis of Classroom Conditions Based on Mathematical Social Science" *Education Sciences* 9, no. 4: 288.
https://doi.org/10.3390/educsci9040288